Split an interval to a sequence of intervals whose union is the
original interval using a given sequence of cut points.
The cut points are expected to be in increasing order and contained
in the given interval. Violations of this rule are tolerated.

a function behaving well on sequences that intersect to a non-empty interval

This produces a function that computes the maximal extension of the
given function. A maximal extension function has the property:
f(I) = { f(x) | x in I }. Here we get this property only for the
limit function for its EffortIndex tending to infinity.